#497509

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#548 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

955 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(6,8)(9,10,14,19,15,20,11,16)(12,23,21,38,25,36,24,40)(13,27,22,37,29,18,30,32)(17,33,34,28,35,39,31,26);(1,2)(3,6)(4,5)(7,8)(9,11,15,14)(10,16,20,19)(12,24,25,21)(13,22,29,30)(17,34,35,31)(18,32,27,37)(23,40,36,38)(26,33,28,39);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,28)(14,24)(15,25)(16,31)(18,36)(19,34)(20,35)(22,33)(23,27)(26,29)(30,39)(32,40)(37,38);(2,5)(6,8)(12,25)(17,35)(21,24)(23,36)(26,28)(31,34)(33,39)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,d^c=di,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404705);